Sequence selection property and bi-Lipschitz homeomorphism
Hyogo University, Japan
In the previous joint paper with Laurentiu Paunescu, we proved that the dimension of the common direction set of two subanalytic subsets is preserved by a bi-Lipschitz homeomorphism, provided that their images are also subanalytic. One of the main ingredients of the proof is the sequence selection property, denoted by (SSP) for short. This property is not invariant under a bi-Lipschitz homeomorphsin, but they are well suited. Our aim is to establish the geometry of sets satisfying (SSP) with bi-Lipschitz tranformations. In this talk we mention some fundamental results on (SSP), e.g. weak transversality theorem in the singular case, (SSP) structure preserving theorem, and so on.